Stars and bars: add positive integer sum theorem

jxu · web-flow · commit 515a2548a790 · 2023-10-19T23:53:07.000-04:00
diff --git a/src/combinatorics/stars_and_bars.md b/src/combinatorics/stars_and_bars.md
@@ -39,6 +39,17 @@ E.g. the solution $1 + 3 + 0 = 4$ for $n = 4$, $k = 3$ can be represented using
 It is easy to see, that this is exactly the stars and bars theorem.
 Therefore the solution is $\binom{n + k - 1}{n}$.
 
+## Number of positive integer sums
+
+A second theorem provides a nice interpretation for positive integers. Consider solutions to 
+
+$$x_1 + x_2 + \dots + x_k = n$$
+
+with $x_i \ge 1$.
+
+We can consider $n$ stars, but this time we can put at most _one bar_ between stars, since two bars between stars would represent $x_i=0$, i.e. an empty box. 
+There are $n-1$ gaps between stars to place $k-1$ bars, so the solution is $\binom{n-1}{k-1}$. 
+
 ## Number of lower-bound integer sums
 
 This can easily be extended to integer sums with different lower bounds.
